Set-theoretic methodology using fuzzy sets in rule extraction and validation - consistency and coverage revisited

- Performance of set-theoretic consistency and coverage is discussed and analyzed.
- Shortcomings of their fuzzification as proposed by Ragin in 2006 are discussed.
- Correction of their fuzzification is proposed and discussed.
- The concept of degree of support and degree of disproof of a rule is proposed.
- Performance of the proposed measures is illustrated graphically and analyzed.

The use of set-theoretic concepts of consistency and coverage and their fuzzified versions needs to be accompanied with an extension of the general (non-fuzzy) methodology. In the fuzzy context, the same piece of data can provide partial evidence in favour of the existence of a given relationship, and at the same time contribute to its disproof, which constitutes a significant interpretation and methodological issue. We point out the possible problems of the use of these measures fuzzified in direct analogy to their non-fuzzy counterparts in the investigation of relationships and causality. We propose two alternative fuzzifications of these measures to be used in the set-theoretic framework and fuzzy set qualitative comparative analysis (fsQCA). We also introduce novel degree-of-support and degree-of-disproof measures to handle simultaneous subset relations. In this way, simultaneous partial support for- and disproof of a given rule by the given set of data can be analyzed in detail. The suggested measures can enhance the insights needed in theory building and be widely used in research using fuzzified set-theoretic methods for the assessment of the plausibility of the rules. We compare the newly suggested consistency and coverage measures with standard ones and discuss their properties in practical examples.